Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.
Factor a quadratic and solve its zero factors
Problem
Factor \(x^{2}-7x+12\) completely by finding integers with product \(12\) and sum \(-7\). Set each factor equal to zero, solve for \(\text{both}~\text{zeros}\), and expand the product to check the original quadratic.
Big Picture
What this problem is really about
Factoring and solving are two distinct stages joined by the zero-product property. We’ll first match a signed integer pair to the constant product and linear sum, verify the factorization by expansion, and only then solve each zero factor.
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